Solving high-dimensional parabolic PDEs using the tensor train format

Lorenz Richter, Leon Sallandt, Nikolas Nüsken
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:8998-9009, 2021.

Abstract

High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the curse of dimensionality. In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs: the combination of reformulations in terms of backward stochastic differential equations and regression-type methods in the tensor format holds the promise of leveraging latent low-rank structures enabling both compression and efficient computation. Following this paradigm, we develop novel iterative schemes, involving either explicit and fast or implicit and accurate updates. We demonstrate in a number of examples that our methods achieve a favorable trade-off between accuracy and computational efficiency in comparison with state-of-the-art neural network based approaches.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-richter21a, title = {Solving high-dimensional parabolic PDEs using the tensor train format}, author = {Richter, Lorenz and Sallandt, Leon and N{\"u}sken, Nikolas}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {8998--9009}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/richter21a/richter21a.pdf}, url = {https://proceedings.mlr.press/v139/richter21a.html}, abstract = {High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the curse of dimensionality. In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs: the combination of reformulations in terms of backward stochastic differential equations and regression-type methods in the tensor format holds the promise of leveraging latent low-rank structures enabling both compression and efficient computation. Following this paradigm, we develop novel iterative schemes, involving either explicit and fast or implicit and accurate updates. We demonstrate in a number of examples that our methods achieve a favorable trade-off between accuracy and computational efficiency in comparison with state-of-the-art neural network based approaches.} }
Endnote
%0 Conference Paper %T Solving high-dimensional parabolic PDEs using the tensor train format %A Lorenz Richter %A Leon Sallandt %A Nikolas Nüsken %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-richter21a %I PMLR %P 8998--9009 %U https://proceedings.mlr.press/v139/richter21a.html %V 139 %X High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the curse of dimensionality. In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs: the combination of reformulations in terms of backward stochastic differential equations and regression-type methods in the tensor format holds the promise of leveraging latent low-rank structures enabling both compression and efficient computation. Following this paradigm, we develop novel iterative schemes, involving either explicit and fast or implicit and accurate updates. We demonstrate in a number of examples that our methods achieve a favorable trade-off between accuracy and computational efficiency in comparison with state-of-the-art neural network based approaches.
APA
Richter, L., Sallandt, L. & Nüsken, N.. (2021). Solving high-dimensional parabolic PDEs using the tensor train format. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:8998-9009 Available from https://proceedings.mlr.press/v139/richter21a.html.

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